Proof strategy After O( Φ -2 log n) steps we have Vol (INFORMED) > Vol (G)/2 w.h.p. E 1 E 2 P(E 1 ) = P(E 2 )
Proof strategy • After O( Φ -2 log n) steps we have Vol (INFORMED) > Vol (G)/2 w.h.p. • After O( Φ -2 log n) steps each node pulls the information from a set of nodes of Vol (G)/2 w.h.p. • After O( Φ -2 log n) steps all the nodes have the info w.h.p.
Key lemma S’ S After O( Φ -1 ) steps with constant probability, we have that for the new set of informed nodes S’ Vol (S’) ≥ (1+ Ω ( Φ )) Vol (S)
Sketch of the proof S Idea: analyze what happens to each node in S in a macro-phase
Sketch of the proof S v 0
Sketch of the proof S v 0
Sketch of the proof S v 0 G ( v 0 )
Sketch of the proof S v 0 v 1 G ( v 0 )
Sketch of the proof S v 0 v 1 G ( v 0 )
Sketch of the proof S v 0 v 1 G ( v 0 )
Sketch of the proof S v 0 v 1 G ( v 0 )
Sketch of the proof S v 0 v 1 G ( v 0 ) G ( v 1 )
Sketch of the proof S We define the following sets:
Sketch of the proof A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider
Sketch of the proof A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider
Sketch of the proof B A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase
Sketch of the proof B A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase
Sketch of the proof B A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase
Sketch of the proof B A S We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase
Sketch of the proof B A S U We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase • U useful nodes in A
Sketch of the proof B A S U We define the following sets: • A ⊆ S , informed nodes that we still have to consider • B ⊇ S, informed nodes at the current phase • U useful nodes in A
Sketch of the proof U B S A The set of useful nodes U is v ∈ A | deg + � � B ( v ) deg( v ) ≥ φ U = U B ( A ) = 2
Sketch of the proof U B S A The set of useful nodes U is v ∈ A | deg + � � B ( v ) deg( v ) ≥ φ U = U B ( A ) = 2 1. The cut (U, V - B) is a large part of the cut (S, V - S), which has size at least Φ Vol (S).
Sketch of the proof U B S A The set of useful nodes U is v ∈ A | deg + � � B ( v ) deg( v ) ≥ φ U = U B ( A ) = 2 1. The cut (U, V - B) is a large part of the cut (S, V - S), which has size at least Φ Vol (S). 2. And, furthermore, each node in U will have constant probability of gaining a constant fraction of its edges in the cut.
Sketch of the proof In order to get the key lemma we prove that for every macro- phase, and every v in U � � G ( v ) ≥ 1 20 · deg + ≥ 1 − e − 1 Pr B ( v )
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